Precalculus Examples

$f (x) = \frac{x^{\frac{1}{3}}}{5} + 8$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{x^{\frac{1}{3}}}{5} + 8$ as an equation.

$y = \frac{x^{\frac{1}{3}}}{5} + 8$

Step 2

Interchange the variables.

$x = \frac{y^{\frac{1}{3}}}{5} + 8$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{y^{\frac{1}{3}}}{5} + 8 = x$ .

$\frac{y^{\frac{1}{3}}}{5} + 8 = x$

Step 3.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.2.1

Subtract $8$ from both sides of the equation.

$\frac{y^{\frac{1}{3}}}{5} - x = - 8$

Step 3.2.2

Add $x$ to both sides of the equation.

$\frac{y^{\frac{1}{3}}}{5} = - 8 + x$

Step 3.3

Multiply both sides of the equation by $5$ .

$5 \frac{y^{\frac{1}{3}}}{5} = 5 (- 8 + x)$

Step 3.4

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.1

Simplify the left side.

Tap for more steps...

Step 3.4.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.4.1.1.1

Cancel the common factor.

$5 \frac{y^{\frac{1}{3}}}{5} = 5 (- 8 + x)$

Step 3.4.1.1.2

Rewrite the expression.

$y^{\frac{1}{3}} = 5 (- 8 + x)$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $5 (- 8 + x)$ .

Tap for more steps...

Step 3.4.2.1.1

Apply the distributive property.

$y^{\frac{1}{3}} = 5 \cdot - 8 + 5 x$

Step 3.4.2.1.2

Multiply $5$ by $- 8$ .

$y^{\frac{1}{3}} = - 40 + 5 x$

Step 3.5

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 40 + 5 x)}^{3}$

Step 3.6

Simplify the exponent.

Tap for more steps...

Step 3.6.1

Simplify the left side.

Tap for more steps...

Step 3.6.1.1

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.6.1.1.1

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.6.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{3} \cdot 3} = {(- 40 + 5 x)}^{3}$

Step 3.6.1.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.6.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 40 + 5 x)}^{3}$

Step 3.6.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(- 40 + 5 x)}^{3}$

Step 3.6.1.1.2

Simplify.

$y = {(- 40 + 5 x)}^{3}$

Step 3.6.2

Simplify the right side.

Tap for more steps...

Step 3.6.2.1

Simplify ${(- 40 + 5 x)}^{3}$ .

Tap for more steps...

Step 3.6.2.1.1

Use the Binomial Theorem.

$y = {(- 40)}^{3} + 3 {(- 40)}^{2} (5 x) + 3 \cdot - 40 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2

Simplify each term.

Tap for more steps...

Step 3.6.2.1.2.1

Raise $- 40$ to the power of $3$ .

$y = - 64000 + 3 {(- 40)}^{2} (5 x) + 3 \cdot - 40 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.2

Raise $- 40$ to the power of $2$ .

$y = - 64000 + 3 \cdot 1600 (5 x) + 3 \cdot - 40 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.3

Multiply $3$ by $1600$ .

$y = - 64000 + 4800 (5 x) + 3 \cdot - 40 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.4

Multiply $5$ by $4800$ .

$y = - 64000 + 24000 x + 3 \cdot - 40 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.5

Multiply $3$ by $- 40$ .

$y = - 64000 + 24000 x - 120 {(5 x)}^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.6

Apply the product rule to $5 x$ .

$y = - 64000 + 24000 x - 120 (5^{2} x^{2}) + {(5 x)}^{3}$

Step 3.6.2.1.2.7

Raise $5$ to the power of $2$ .

$y = - 64000 + 24000 x - 120 (25 x^{2}) + {(5 x)}^{3}$

Step 3.6.2.1.2.8

Multiply $25$ by $- 120$ .

$y = - 64000 + 24000 x - 3000 x^{2} + {(5 x)}^{3}$

Step 3.6.2.1.2.9

Apply the product rule to $5 x$ .

$y = - 64000 + 24000 x - 3000 x^{2} + 5^{3} x^{3}$

Step 3.6.2.1.2.10

Raise $5$ to the power of $3$ .

$y = - 64000 + 24000 x - 3000 x^{2} + 125 x^{3}$

Step 3.7

Simplify $- 64000 + 24000 x - 3000 x^{2} + 125 x^{3}$ .

Tap for more steps...

Step 3.7.1

Move $- 64000$ .

$y = 24000 x - 3000 x^{2} + 125 x^{3} - 64000$

Step 3.7.2

Move $24000 x$ .

$y = - 3000 x^{2} + 125 x^{3} + 24000 x - 64000$

Step 3.7.3

Reorder $- 3000 x^{2}$ and $125 x^{3}$ .

$y = 125 x^{3} - 3000 x^{2} + 24000 x - 64000$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 125 x^{3} - 3000 x^{2} + 24000 x - 64000$

Step 5

Verify if $f^{- 1} (x) = 125 x^{3} - 3000 x^{2} + 24000 x - 64000$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{5} + 8$ .

Tap for more steps...

Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{3} - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 ({(\frac{x^{\frac{1}{3}}}{5})}^{3} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2

Simplify each term.

Tap for more steps...

Step 5.2.3.2.1

Apply the product rule to $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{{(x^{\frac{1}{3}})}^{3}}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.2

Simplify the numerator.

Tap for more steps...

Step 5.2.3.2.2.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 5.2.3.2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x^{\frac{1}{3} \cdot 3}}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.2.2.1.2.1

Cancel the common factor.

Step 5.2.3.2.2.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{5^{3}} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.2.2

Simplify.

Step 5.2.3.2.3

Raise $5$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + 3 {(\frac{x^{\frac{1}{3}}}{5})}^{2} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.4

Apply the product rule to $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + 3 (\frac{{(x^{\frac{1}{3}})}^{2}}{5^{2}}) \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

Tap for more steps...

Step 5.2.3.2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{1}{3} \cdot 2}}{5^{2}}) \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.5.2

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{2}{3}}}{5^{2}}) \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.6

Raise $5$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + 3 (\frac{x^{\frac{2}{3}}}{25}) \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.7

Combine $3$ and $\frac{x^{\frac{2}{3}}}{25}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{3 x^{\frac{2}{3}}}{25} \cdot 8 + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.8

Multiply $\frac{3 x^{\frac{2}{3}}}{25} \cdot 8$ .

Tap for more steps...

Step 5.2.3.2.8.1

Combine $\frac{3 x^{\frac{2}{3}}}{25}$ and $8$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{3 x^{\frac{2}{3}} \cdot 8}{25} + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.8.2

Multiply $8$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + 3 (\frac{x^{\frac{1}{3}}}{5}) \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.9

Combine $3$ and $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}}}{5} \cdot 8^{2} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.10

Raise $8$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}}}{5} \cdot 64 + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.11

Multiply $\frac{3 x^{\frac{1}{3}}}{5} \cdot 64$ .

Tap for more steps...

Step 5.2.3.2.11.1

Combine $\frac{3 x^{\frac{1}{3}}}{5}$ and $64$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + \frac{3 x^{\frac{1}{3}} \cdot 64}{5} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.11.2

Multiply $64$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + \frac{192 x^{\frac{1}{3}}}{5} + 8^{3}) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.2.12

Raise $8$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125} + \frac{24 x^{\frac{2}{3}}}{25} + \frac{192 x^{\frac{1}{3}}}{5} + 512) - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.3

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = 125 (\frac{x}{125}) + 125 (\frac{24 x^{\frac{2}{3}}}{25}) + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4

Simplify.

Tap for more steps...

Step 5.2.3.4.1

Cancel the common factor of $125$ .

Tap for more steps...

Step 5.2.3.4.1.1

Cancel the common factor.

Step 5.2.3.4.1.2

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 125 (\frac{24 x^{\frac{2}{3}}}{25}) + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.2

Cancel the common factor of $25$ .

Tap for more steps...

Step 5.2.3.4.2.1

Factor $25$ out of $125$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 25 (5) (\frac{24 x^{\frac{2}{3}}}{25}) + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.2.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 25 \cdot (5 (\frac{24 x^{\frac{2}{3}}}{25})) + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.2.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 5 (24 x^{\frac{2}{3}}) + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.3

Multiply $24$ by $5$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 125 (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.4.4.1

Factor $5$ out of $125$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 5 (25) (\frac{192 x^{\frac{1}{3}}}{5}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.4.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 5 \cdot (25 (\frac{192 x^{\frac{1}{3}}}{5})) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.4.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 25 (192 x^{\frac{1}{3}}) + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.5

Multiply $192$ by $25$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 125 \cdot 512 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.4.6

Multiply $125$ by $512$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 {(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2} + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.5

Rewrite ${(\frac{x^{\frac{1}{3}}}{5} + 8)}^{2}$ as $(\frac{x^{\frac{1}{3}}}{5} + 8) (\frac{x^{\frac{1}{3}}}{5} + 8)$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 ((\frac{x^{\frac{1}{3}}}{5} + 8) (\frac{x^{\frac{1}{3}}}{5} + 8)) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.6

Expand $(\frac{x^{\frac{1}{3}}}{5} + 8) (\frac{x^{\frac{1}{3}}}{5} + 8)$ using the FOIL Method.

Tap for more steps...

Step 5.2.3.6.1

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1}{3}}}{5} \cdot (\frac{x^{\frac{1}{3}}}{5} + 8) + 8 (\frac{x^{\frac{1}{3}}}{5} + 8)) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.6.2

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1}{3}}}{5} \cdot \frac{x^{\frac{1}{3}}}{5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5} + 8)) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.6.3

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1}{3}}}{5} \cdot \frac{x^{\frac{1}{3}}}{5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7

Simplify and combine like terms.

Tap for more steps...

Step 5.2.3.7.1

Simplify each term.

Tap for more steps...

Step 5.2.3.7.1.1

Combine.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1}{3}} x^{\frac{1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.2

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 5.2.3.7.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1}{3} + \frac{1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.2.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{1 + 1}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.2.3

Add $1$ and $1$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{5 \cdot 5} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.3

Multiply $5$ by $5$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{x^{\frac{1}{3}}}{5} \cdot 8 + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.4

Combine $\frac{x^{\frac{1}{3}}}{5}$ and $8$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{x^{\frac{1}{3}} \cdot 8}{5} + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.5

Move $8$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{8 \cdot x^{\frac{1}{3}}}{5} + 8 (\frac{x^{\frac{1}{3}}}{5}) + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.6

Combine $8$ and $\frac{x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{8 x^{\frac{1}{3}}}{5} + \frac{8 x^{\frac{1}{3}}}{5} + 8 \cdot 8) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.1.7

Multiply $8$ by $8$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{8 x^{\frac{1}{3}}}{5} + \frac{8 x^{\frac{1}{3}}}{5} + 64) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.7.2

Add $\frac{8 x^{\frac{1}{3}}}{5}$ and $\frac{8 x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + 2 (\frac{8 x^{\frac{1}{3}}}{5}) + 64) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.8

Multiply $2 \frac{8 x^{\frac{1}{3}}}{5}$ .

Tap for more steps...

Step 5.2.3.8.1

Combine $2$ and $\frac{8 x^{\frac{1}{3}}}{5}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{2 (8 x^{\frac{1}{3}})}{5} + 64) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.8.2

Multiply $8$ by $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 (\frac{x^{\frac{2}{3}}}{25} + \frac{16 x^{\frac{1}{3}}}{5} + 64) + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.9

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 3000 \frac{x^{\frac{2}{3}}}{25} - 3000 \frac{16 x^{\frac{1}{3}}}{5} - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10

Simplify.

Tap for more steps...

Step 5.2.3.10.1

Cancel the common factor of $25$ .

Tap for more steps...

Step 5.2.3.10.1.1

Factor $25$ out of $- 3000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 + 25 (- 120) (\frac{x^{\frac{2}{3}}}{25}) - 3000 \frac{16 x^{\frac{1}{3}}}{5} - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.1.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 + 25 \cdot (- 120 \frac{x^{\frac{2}{3}}}{25}) - 3000 \frac{16 x^{\frac{1}{3}}}{5} - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.1.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 3000 \frac{16 x^{\frac{1}{3}}}{5} - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.10.2.1

Factor $5$ out of $- 3000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} + 5 (- 600) (\frac{16 x^{\frac{1}{3}}}{5}) - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.2.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} + 5 \cdot (- 600 \frac{16 x^{\frac{1}{3}}}{5}) - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.2.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 600 (16 x^{\frac{1}{3}}) - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.3

Multiply $16$ by $- 600$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 3000 \cdot 64 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.10.4

Multiply $- 3000$ by $64$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 24000 (\frac{x^{\frac{1}{3}}}{5} + 8) - 64000$

Step 5.2.3.11

Apply the distributive property.

Step 5.2.3.12

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.12.1

Factor $5$ out of $24000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 5 (4800) (\frac{x^{\frac{1}{3}}}{5}) + 24000 \cdot 8 - 64000$

Step 5.2.3.12.2

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 5 \cdot (4800 (\frac{x^{\frac{1}{3}}}{5})) + 24000 \cdot 8 - 64000$

Step 5.2.3.12.3

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 4800 x^{\frac{1}{3}} + 24000 \cdot 8 - 64000$

Step 5.2.3.13

Multiply $24000$ by $8$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 4800 x^{\frac{1}{3}} + 192000 - 64000$

Step 5.2.4

Simplify by adding terms.

Tap for more steps...

Step 5.2.4.1

Combine the opposite terms in $x + 120 x^{\frac{2}{3}} + 4800 x^{\frac{1}{3}} + 64000 - 120 x^{\frac{2}{3}} - 9600 x^{\frac{1}{3}} - 192000 + 4800 x^{\frac{1}{3}} + 192000 - 64000$ .

Tap for more steps...

Step 5.2.4.1.1

Subtract $120 x^{\frac{2}{3}}$ from $120 x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} + 64000 + 0 - 9600 x^{\frac{1}{3}} - 192000 + 4800 x^{\frac{1}{3}} + 192000 - 64000$

Step 5.2.4.1.2

Add $x + 4800 x^{\frac{1}{3}} + 64000$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} + 64000 - 9600 x^{\frac{1}{3}} - 192000 + 4800 x^{\frac{1}{3}} + 192000 - 64000$

Step 5.2.4.1.3

Add $- 192000$ and $192000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} + 64000 - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}} + 0 - 64000$

Step 5.2.4.1.4

Add $x + 4800 x^{\frac{1}{3}} + 64000 - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} + 64000 - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}} - 64000$

Step 5.2.4.1.5

Subtract $64000$ from $64000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}} + 0$

Step 5.2.4.1.6

Add $x + 4800 x^{\frac{1}{3}} - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 4800 x^{\frac{1}{3}} - 9600 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}}$

Step 5.2.4.2

Subtract $9600 x^{\frac{1}{3}}$ from $4800 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x - 4800 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}}$

Step 5.2.4.3

Combine the opposite terms in $x - 4800 x^{\frac{1}{3}} + 4800 x^{\frac{1}{3}}$ .

Tap for more steps...

Step 5.2.4.3.1

Add $- 4800 x^{\frac{1}{3}}$ and $4800 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{5} + 8) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (125 x^{3} - 3000 x^{2} + 24000 x - 64000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (125 x^{3} - 3000 x^{2} + 24000 x - 64000) = \frac{{(125 x^{3} - 3000 x^{2} + 24000 x - 64000)}^{\frac{1}{3}}}{5} + 8$

Step 5.3.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (125 x^{3} - 3000 x^{2} + 24000 x - 64000) = \frac{{(125 x^{3} - 3000 x^{2} + 24000 x - 64000)}^{\frac{1}{3}}}{5} + 8 \cdot \frac{5}{5}$

Step 5.3.4

Combine $8$ and $\frac{5}{5}$ .

$f (125 x^{3} - 3000 x^{2} + 24000 x - 64000) = \frac{{(125 x^{3} - 3000 x^{2} + 24000 x - 64000)}^{\frac{1}{3}}}{5} + \frac{8 \cdot 5}{5}$

Step 5.3.5

Combine the numerators over the common denominator.

$f (125 x^{3} - 3000 x^{2} + 24000 x - 64000) = \frac{{(125 x^{3} - 3000 x^{2} + 24000 x - 64000)}^{\frac{1}{3}} + 8 \cdot 5}{5}$

Step 5.3.6

Multiply $8$ by $5$ .

$f (125 x^{3} - 3000 x^{2} + 24000 x - 64000) = \frac{{(125 x^{3} - 3000 x^{2} + 24000 x - 64000)}^{\frac{1}{3}} + 40}{5}$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 125 x^{3} - 3000 x^{2} + 24000 x - 64000$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{5} + 8$ .

$f^{- 1} (x) = 125 x^{3} - 3000 x^{2} + 24000 x - 64000$